Abstract
Spin blockade occurs when an electron is unable to access an energetically favourable path through a quantum dot owing to spin conservation, resulting in a blockade of the current through the dot^{1,2,3,4,5,6}. Spin blockade is the basis of a number of recent advances in spintronics, including the measurement and the manipulation of individual electron spins^{7,8}. We report measurements of the spin blockade regime in a silicon double quantum dot, revealing a complementary phenomenon: lifetimeenhanced transport. We argue that our observations arise because the decay times for electron spins in silicon are long, enabling the electron to maintain its spin throughout its transit across the quantum dot and access fast paths that exist in some spin channels but not in others. Such long spin lifetimes are important for applications such as quantum computation and, more generally, spintronics.
Main
Semiconductor quantum dots or ‘artificial atoms’ provide highly tunable structures for trapping and manipulating individual electrons^{9,10,11}. Such quantum dots are promising candidates as qubits for quantum computation^{12,13,14}, owing in part to the long lifetimes and slow dephasing of electron spins in semiconductors^{7,15}. Si quantum dots are predicted to have especially long lifetimes and slow dephasing, due to low spin–orbit interaction and low nuclear spin density^{16,17}. In the past several years, much activity has focused on the development of quantum dots in Si/SiGe (refs 18, 19, 20, 21, 22) and recent advances in materials quality and fabrication techniques have enabled the observation of coherent spin phenomena in such quantum dots^{23}.
Spintocharge conversion, in which spin states are detected through their effect on charge motion, enables measurement of individual electron spins in quantum dots^{15}. Spin blockade is the canonical example of spintocharge conversion in transport, where charge current is blocked in a double quantum dot by a metastable spin state. The blockade occurs when one electron is confined in the left dot and a further electron enters the right dot forming a spin triplet state T(1,1) (Fig. 1a). Exiting the dot requires reaching the triplet T(2,0), with both electrons in the left dot, a state that is higher in energy. The electron is thus trapped in the right dot, unless relaxation from T(1,1) to S(1,1) occurs, opening a downhill channel through S(2,0). As we show below, this aspect of spin blockade in Si is virtually identical to that previously observed in other systems^{1,2,3}.
The unexpected effect presented here is lifetimeenhanced transport (LET). The energy level diagram for LET is the same as for spin blockade, except that current flows in the opposite direction (Fig. 1b). Flow through the triplet channel is now energetically downhill, whereas flow through the singlet channel is very slow, because it requires either an uphill transition or tunnelling directly from the left dot to the right lead. Transport current will be observable only if electrons flow almost exclusively through the triplet channel, requiring even slower triplet–singlet relaxation rates than those needed to observe spin blockade.
The tunable quantum dot used in these experiments was formed in a Si/SiGe heterostructure. The gate structure (Fig. 2a) has the shape often associated with a single quantum dot, and the corresponding Coulomb diamonds are shown in Fig. 2b. By tuning the gate voltages, the single dot was split into two tunnelcoupled quantum dots. Such transformations of a lateral single quantum dot into multiple quantum dots have been demonstrated in similar systems^{24,25}. Here, by changing voltages on gates G and CS, and keeping those on B_{L}, T and B_{R} fixed, the electron occupations are tuned while keeping the tunnel barriers constant (Fig. 2d). The left dot is coupled more strongly to gate G, and the right dot is coupled more strongly to gate CS. The electron occupancies indicated in the figure correspond to an equivalent charge configuration with a single unpaired spin in the (1,0) state.
The region of interest here is indicated by the blue dashed circle in Fig. 2d. The lower of these ‘triple points’ corresponds to degeneracy between the (1,0), (1,1) and (2,0) charge states^{26,27}. In this regime, an electron with a spin ↑〉 or ↓〉 is confined in the left dot, and the incoming electron can form either a spin singlet or any of the spin triplets T_{+}:↑↑〉, T_{−}:↓↓〉 or , which are degenerate at zero magnetic field. The singlet–triplet energy splitting is larger for two electrons occupying the same dot (2,0), than when they are in different dots (1,1), resulting in the energy level schematic diagrams shown in Fig. 1.
Spin blockade arises because spin is conserved during tunnelling, preventing the direct transition from the triplet T(1,1) to the singlet S(2,0). We observe this blockade as shown in Fig. 3a–c. These measurements are taken at finite bias, where the triple points expand into bias triangles^{28}. When T(1,1) is loaded and no relaxation occurs from T(1,1) to S(1,1), spin blockade is observed (as marked by the orange triangle) and the bias triangle is truncated as shown schematically in Fig. 3b. The observed current in the blockaded region is limited by the noise floor in the measurement (7 fA r.m.s.). Spin blockade is fully lifted when the T(2,0) state is brought below the T(1,1) state (blue star).
Now consider the same energy level configuration, but with opposite bias across the dot (Fig. 1b). In previous work, this configuration has been shown to be blockaded^{2,8}. In contrast, here in Si we observe a strong ‘tail’ of current in this configuration, corresponding to the extra parallelograms (green outline) in Fig. 3d,e. As shown in detail below, the condition for observing this tail is that the metastable S(2,0) state must be loaded more slowly than it empties. The relaxation rate from the T(2,0) state into S(2,0) sets a lower bound for this loading rate. Because the measured current at the point labelled (+) is significant only when the spin lifetime of T(2,0) is long, we denote this tail of current the triplet tail and the effect LET.
The dimensions of the triplet tail in the charge stability diagram (Fig. 3d,e) provide a measurement of the energy difference between the (2,0) triplet and singlet states (E_{ST}=E_{T}−E_{S}). Both the length of the tail and the distance between the tail and the edge of the bias triangle correspond to E_{ST} (Fig. 3e). This (2,0) singlet–triplet energy gap as extracted from the data is 240±30 μeV.
A simple rate model gives insight into when LET occurs. The rates in the model correspond to transitions between the states shown in Fig. 1b, and the corresponding lifetimes are the inverses of the rates. By calculating the expected amount of time required for an electron to pass through the system, we obtain a quantity proportional to the measured current I (see Supplementary Information for the complete analysis). The slow rates are of interest here: the relaxation rate Γ_{TS} from the triplet T(2,0) to the singlet S(2,0), the loading rate Γ_{LS} of the singlet S(2,0) from the lead and the unloading rate Γ_{S} of the singlet S(2,0). To focus on these rates and to develop intuition, we assume that all other rates are equal to a single rate, Γ_{fast}, an assumption that does not change the qualitative understanding. The resulting proportionality for the current is
As this proportionality shows, the triplet tail is observed if and only if the sum of the triplet–singlet relaxation rate Γ_{TS} and the loading rate from the lead Γ_{LS} is not large compared with the escape rate Γ_{S}. If the triplet–singlet relaxation rate is much faster than the escape rate, then the tail regime will be blockaded by electrons trapped in the S(2,0) state. In our experiments, essentially no reduction in current (∼5%) is observed moving from the bias triangle into the tail (from the blue diamond towards the teal cross in Fig. 3d). Thus, electrons are rarely trapped in S(2,0), indicating that the triplet–singlet relaxation rate Γ_{TS} and the loading rate from the left lead Γ_{LS} are both much less than Γ_{S}, itself a slow rate. A similar calculation with the opposite bias shows that the condition for spin blockade is that the triplet–singlet relaxation rate in the (1,1) configuration is much slower than the fast rates, a far less stringent condition.
To investigate the rapid tunnelling between dots 1 and 2, and to understand the device physics, we have modelled the device numerically, as shown in Fig. 2c. Established methods are used to treat the various charge regions selfconsistently, including trapped surface charge, ionized dopants, the twodimensional electron gas (2DEG) and the device^{29}. The dopants are treated in the jellium approximation, whereas the inhomogeneous depletion of the 2DEG is treated semiclassically in a 2D Thomas–Fermi approximation. For the gated region, a 2D Hartree–Fock basis of singleelectron orbitals is obtained from the effective mass envelope equation, and a twoelectron singlet wavefunction is constructed using the configuration interaction method, similar to ref. 14. The results show that the bottom of the quantum dot confinement potential is nearly flat, with an oblong shape about 200 nm across. General arguments suggest that the electron–electron interactions should dominate the kinetic energy in silicon for electrons separated by over 100 nm, causing two electrons to form a double dot. The modelling results, shown in Fig. 2c, confirm that these general arguments give the correct intuition. As is clear from the figure, the effective tunnel barrier between the two dots is low, consistent with LET. We note that quantum dot splitting has been observed elsewhere, where it was attributed to deformation by a gate potential^{24} or a local impurity^{25}. Although inhomogeneous confinement may also be present in our device, it is not needed to explain the double dot.
LET should be observable in many materials systems, provided the appropriate ratio of rates can be obtained. Indeed, slow triplet–singlet relaxation and the preferential loading of triplets versus singlets have both been observed in GaAs quantum dots, in pulsedgate experiments^{30}. By analysing the current versus voltage data in our bias triangles (see the Supplementary Information), we find that in the tail regime, triplet loading occurs at a rate at least 1,000 times greater than singlet loading. This ratio is 50 times greater than in previous observations of spindependent tunnelling^{30}, which may lead to corresponding enhancements in spin readout. The singlet loading is suppressed because its tunnel barrier to the external lead is larger than that of the triplet state^{31}; the relatively large effective mass of Si enhances this suppression. The higher unloading rate is a consequence of the relatively small tunnel barrier between the two dots, as confirmed by numerical modelling. Our bound on the singlet loading rate places a weak bound on triplet–singlet relaxation of Γ_{TS}<63,000 s^{−1}, although the actual value is expected to be much smaller^{16}.
The phenomena described above of spin blockade and its complementary effect of LET can be unified by measurements of the system with an applied inplane magnetic field. In a magnetic field, the spin triplets are split linearly by the Zeeman energy (E_{Z}=g μ_{B}B S_{Z}), where μ_{B} is the Bohr magneton, B is the magnetic field and S_{Z} is +1 for ↑↑〉, −1 for ↓↓〉 and 0 for . As the gfactor is positive for silicon, the T_{−} states shift lower in energy compared with the T_{0} states, providing a technique for testing the interpretation of the data proposed above.
Figure 4a–c (d–f) shows the energy diagrams for the cases where E_{Z} is less (more) than E_{ST}. In Fig. 4a–c, the ground state of the (1,1) configuration is T_{−}(1,1), whereas that of the (2,0) configuration is S(2,0). Spin blockade now occurs in a smaller region than at B=0, as indicated by dashed lines and the orange circle in Fig. 4a. Spin blockade is lifted in the conventional way when T_{−}(2,0) is lowered below T_{−}(1,1), corresponding to the triangular regions on the lower right in Fig. 4a,b. Spin blockade is also lifted when the S(1,1) state can participate in transport (blue diamond). However, the lifting of the blockade in this case is due to LET, because this S(1,1) state is an excited state of the (1,1) configuration, giving rise to a singlet tail in Fig. 4a,b. This tail is a striking example of a generalization of LET: the singlet–triplet splitting is now inverted, and the LET is now due to long lifetimes in the singlet channel rather than the triplet channel. LET can be generalized to any situation where electron transport occurs through longlived excited states, whereas lower energy states that would be metastably trapped are avoided.
When E_{Z}>E_{ST} (Fig. 4d–f), the ground state configurations are T_{−}(1,1) and T_{−}(2,0), and there should be no spin blockade because groundstate transitions are allowed (purple diamond). Our measurements indeed show full triangles with no blockade. From the features visible inside the triangles (bright lines parallel to base), various excited states can be identified. LET also occurs through excited spin singlet states in this configuration, giving rise to a tail (green star). These data demonstrate the existence of longlived electron spin states even in the presence of a finite magnetic field, a requirement for various quantum operations.
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Acknowledgements
We thank Nick Leaf for help with the data acquisition programs. Support for this work was provided by NSA, LPS and ARO under contract number W911NF0410389, by the National Science Foundation through DMR0325634 and DMR0520527, and by DOE through DEFG0203ER46028.
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Experimental work carried out by N.S., C.B.S., M.T., L.J.K., H.Q., H.L., D.E.S. and M.A.E. Data analysis carried out by N.S., C.B.S., A.J.R., R.J., M.F., R.H.B., S.N.C. and M.A.E. Project planning carried out by N.S., M.A.E., M.G.L., R.H.B., S.N.C., M.F., R.J. and D.E.S.
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Supplementary Information and Supplementary Figures 1–2 (PDF 307 kb)
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Shaji, N., Simmons, C., Thalakulam, M. et al. Spin blockade and lifetimeenhanced transport in a fewelectron Si/SiGe double quantum dot. Nature Phys 4, 540–544 (2008). https://doi.org/10.1038/nphys988
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DOI: https://doi.org/10.1038/nphys988
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